3.2468 \(\int \frac{1}{(d+e x)^{7/2} \sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=629 \[ -\frac{8 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 e \sqrt{b^2-4 a c}}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )}{15 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}-\frac{2 e \sqrt{a+b x+c x^2} \left (-c e (9 a e+23 b d)+8 b^2 e^2+23 c^2 d^2\right )}{15 \sqrt{d+e x} \left (a e^2-b d e+c d^2\right )^3}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c e (9 a e+23 b d)+8 b^2 e^2+23 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{15 \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^3 \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{8 e \sqrt{a+b x+c x^2} (2 c d-b e)}{15 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )^2}-\frac{2 e \sqrt{a+b x+c x^2}}{5 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )} \]

[Out]

(-2*e*Sqrt[a + b*x + c*x^2])/(5*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(5/2)) - (8*e*(2*c*d - b*e)*Sqrt[a + b*x + c
*x^2])/(15*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(3/2)) - (2*e*(23*c^2*d^2 + 8*b^2*e^2 - c*e*(23*b*d + 9*a*e))*S
qrt[a + b*x + c*x^2])/(15*(c*d^2 - b*d*e + a*e^2)^3*Sqrt[d + e*x]) + (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(23*c^2*d^2 +
8*b^2*e^2 - c*e*(23*b*d + 9*a*e))*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[
Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[
b^2 - 4*a*c])*e)])/(15*(c*d^2 - b*d*e + a*e^2)^3*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[
a + b*x + c*x^2]) - (8*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a
*c])*e)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sq
rt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(15*(c*d^2 - b*d*e +
a*e^2)^2*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.686662, antiderivative size = 629, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {744, 834, 843, 718, 424, 419} \[ -\frac{2 e \sqrt{a+b x+c x^2} \left (-c e (9 a e+23 b d)+8 b^2 e^2+23 c^2 d^2\right )}{15 \sqrt{d+e x} \left (a e^2-b d e+c d^2\right )^3}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c e (9 a e+23 b d)+8 b^2 e^2+23 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{15 \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^3 \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{8 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{15 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}-\frac{8 e \sqrt{a+b x+c x^2} (2 c d-b e)}{15 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )^2}-\frac{2 e \sqrt{a+b x+c x^2}}{5 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^(7/2)*Sqrt[a + b*x + c*x^2]),x]

[Out]

(-2*e*Sqrt[a + b*x + c*x^2])/(5*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(5/2)) - (8*e*(2*c*d - b*e)*Sqrt[a + b*x + c
*x^2])/(15*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(3/2)) - (2*e*(23*c^2*d^2 + 8*b^2*e^2 - c*e*(23*b*d + 9*a*e))*S
qrt[a + b*x + c*x^2])/(15*(c*d^2 - b*d*e + a*e^2)^3*Sqrt[d + e*x]) + (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(23*c^2*d^2 +
8*b^2*e^2 - c*e*(23*b*d + 9*a*e))*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[
Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[
b^2 - 4*a*c])*e)])/(15*(c*d^2 - b*d*e + a*e^2)^3*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[
a + b*x + c*x^2]) - (8*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a
*c])*e)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sq
rt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(15*(c*d^2 - b*d*e +
a*e^2)^2*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])

Rule 744

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1)
*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)),
Int[(d + e*x)^(m + 1)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, x]*(a + b*x + c*x^2)^p, x], x]
/; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e
, 0] && NeQ[m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ
[p]) || ILtQ[Simplify[m + 2*p + 3], 0])

Rule 834

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rule 718

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(2*Rt[b^2 - 4*a*c, 2]
*(d + e*x)^m*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))])/(c*Sqrt[a + b*x + c*x^2]*((2*c*(d + e*x))/(2*c*d -
b*e - e*Rt[b^2 - 4*a*c, 2]))^m), Subst[Int[(1 + (2*e*Rt[b^2 - 4*a*c, 2]*x^2)/(2*c*d - b*e - e*Rt[b^2 - 4*a*c,
2]))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b
, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4]

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rubi steps

\begin{align*} \int \frac{1}{(d+e x)^{7/2} \sqrt{a+b x+c x^2}} \, dx &=-\frac{2 e \sqrt{a+b x+c x^2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac{2 \int \frac{\frac{1}{2} (-5 c d+4 b e)+\frac{3 c e x}{2}}{(d+e x)^{5/2} \sqrt{a+b x+c x^2}} \, dx}{5 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{2 e \sqrt{a+b x+c x^2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac{8 e (2 c d-b e) \sqrt{a+b x+c x^2}}{15 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}+\frac{4 \int \frac{\frac{1}{4} \left (15 c^2 d^2+8 b^2 e^2-c e (19 b d+9 a e)\right )-c e (2 c d-b e) x}{(d+e x)^{3/2} \sqrt{a+b x+c x^2}} \, dx}{15 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{2 e \sqrt{a+b x+c x^2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac{8 e (2 c d-b e) \sqrt{a+b x+c x^2}}{15 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}-\frac{2 e \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right ) \sqrt{a+b x+c x^2}}{15 \left (c d^2-b d e+a e^2\right )^3 \sqrt{d+e x}}-\frac{8 \int \frac{-\frac{1}{8} c \left (15 c^2 d^3+4 b e^2 (b d+a e)-c d e (11 b d+17 a e)\right )-\frac{1}{8} c e \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right ) x}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{15 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac{2 e \sqrt{a+b x+c x^2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac{8 e (2 c d-b e) \sqrt{a+b x+c x^2}}{15 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}-\frac{2 e \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right ) \sqrt{a+b x+c x^2}}{15 \left (c d^2-b d e+a e^2\right )^3 \sqrt{d+e x}}-\frac{(4 c (2 c d-b e)) \int \frac{1}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{15 \left (c d^2-b d e+a e^2\right )^2}+\frac{\left (c \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+b x+c x^2}} \, dx}{15 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac{2 e \sqrt{a+b x+c x^2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac{8 e (2 c d-b e) \sqrt{a+b x+c x^2}}{15 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}-\frac{2 e \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right ) \sqrt{a+b x+c x^2}}{15 \left (c d^2-b d e+a e^2\right )^3 \sqrt{d+e x}}+\frac{\left (\sqrt{2} \sqrt{b^2-4 a c} \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 \sqrt{b^2-4 a c} e x^2}{2 c d-b e-\sqrt{b^2-4 a c} e}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )}{15 \left (c d^2-b d e+a e^2\right )^3 \sqrt{\frac{c (d+e x)}{2 c d-b e-\sqrt{b^2-4 a c} e}} \sqrt{a+b x+c x^2}}-\frac{\left (8 \sqrt{2} \sqrt{b^2-4 a c} (2 c d-b e) \sqrt{\frac{c (d+e x)}{2 c d-b e-\sqrt{b^2-4 a c} e}} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 \sqrt{b^2-4 a c} e x^2}{2 c d-b e-\sqrt{b^2-4 a c} e}}} \, dx,x,\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )}{15 \left (c d^2-b d e+a e^2\right )^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ &=-\frac{2 e \sqrt{a+b x+c x^2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac{8 e (2 c d-b e) \sqrt{a+b x+c x^2}}{15 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}-\frac{2 e \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right ) \sqrt{a+b x+c x^2}}{15 \left (c d^2-b d e+a e^2\right )^3 \sqrt{d+e x}}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{15 \left (c d^2-b d e+a e^2\right )^3 \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{a+b x+c x^2}}-\frac{8 \sqrt{2} \sqrt{b^2-4 a c} (2 c d-b e) \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{15 \left (c d^2-b d e+a e^2\right )^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 9.09226, size = 983, normalized size = 1.56 \[ \frac{2 \sqrt{c x^2+b x+a} \left (\left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right ) \left (c \left (\frac{d}{d+e x}-1\right )^2+\frac{e \left (-\frac{d b}{d+e x}+b+\frac{a e}{d+e x}\right )}{d+e x}\right )-\frac{i \sqrt{1-\frac{2 \left (c d^2+e (a e-b d)\right )}{\left (2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt{\frac{2 \left (c d^2+e (a e-b d)\right )}{\left (-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}+1} \left (\left (2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right ) \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}}}{\sqrt{d+e x}}\right )|-\frac{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right )+\left (-30 c^3 d^3-c^2 \left (-34 a e^2-45 b d e+23 d \sqrt{\left (b^2-4 a c\right ) e^2}\right ) d+8 b^2 e^2 \left (b e-\sqrt{\left (b^2-4 a c\right ) e^2}\right )+c e \left (-31 d e b^2-17 a e^2 b+23 d \sqrt{\left (b^2-4 a c\right ) e^2} b+9 a e \sqrt{\left (b^2-4 a c\right ) e^2}\right )\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}}}{\sqrt{d+e x}}\right ),-\frac{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right )\right )}{2 \sqrt{2} \sqrt{\frac{c d^2+e (a e-b d)}{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}} \sqrt{d+e x}}\right ) (d+e x)^{3/2}}{15 e \left (c d^2-b e d+a e^2\right )^3 \sqrt{a+x (b+c x)} \sqrt{\frac{(d+e x)^2 \left (c \left (\frac{d}{d+e x}-1\right )^2+\frac{e \left (-\frac{d b}{d+e x}+b+\frac{a e}{d+e x}\right )}{d+e x}\right )}{e^2}}}+\frac{\left (c x^2+b x+a\right ) \left (\frac{2 \left (-23 c^2 d^2+23 b c e d-8 b^2 e^2+9 a c e^2\right ) e}{15 \left (c d^2-b e d+a e^2\right )^3 (d+e x)}+\frac{8 (b e-2 c d) e}{15 \left (c d^2-b e d+a e^2\right )^2 (d+e x)^2}-\frac{2 e}{5 \left (c d^2-b e d+a e^2\right ) (d+e x)^3}\right ) \sqrt{d+e x}}{\sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(7/2)*Sqrt[a + b*x + c*x^2]),x]

[Out]

(Sqrt[d + e*x]*(a + b*x + c*x^2)*((-2*e)/(5*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^3) + (8*e*(-2*c*d + b*e))/(15*(c
*d^2 - b*d*e + a*e^2)^2*(d + e*x)^2) + (2*e*(-23*c^2*d^2 + 23*b*c*d*e - 8*b^2*e^2 + 9*a*c*e^2))/(15*(c*d^2 - b
*d*e + a*e^2)^3*(d + e*x))))/Sqrt[a + x*(b + c*x)] + (2*(d + e*x)^(3/2)*Sqrt[a + b*x + c*x^2]*((23*c^2*d^2 + 8
*b^2*e^2 - c*e*(23*b*d + 9*a*e))*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*
x)) - ((I/2)*Sqrt[1 - (2*(c*d^2 + e*(-(b*d) + a*e)))/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt
[1 + (2*(c*d^2 + e*(-(b*d) + a*e)))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*((2*c*d - b*e + Sqrt
[(b^2 - 4*a*c)*e^2])*(23*c^2*d^2 + 8*b^2*e^2 - c*e*(23*b*d + 9*a*e))*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2
- b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*
c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))] + (-30*c^3*d^3 + 8*b^2*e^2*(b*e - Sqrt[(b^2 - 4*a*c)*e^2])
- c^2*d*(-45*b*d*e - 34*a*e^2 + 23*d*Sqrt[(b^2 - 4*a*c)*e^2]) + c*e*(-31*b^2*d*e - 17*a*b*e^2 + 23*b*d*Sqrt[(b
^2 - 4*a*c)*e^2] + 9*a*e*Sqrt[(b^2 - 4*a*c)*e^2]))*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(
-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d -
b*e + Sqrt[(b^2 - 4*a*c)*e^2]))]))/(Sqrt[2]*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)
*e^2])]*Sqrt[d + e*x])))/(15*e*(c*d^2 - b*d*e + a*e^2)^3*Sqrt[a + x*(b + c*x)]*Sqrt[((d + e*x)^2*(c*(-1 + d/(d
+ e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x)))/e^2])

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Maple [B]  time = 0.412, size = 14311, normalized size = 22.8 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(7/2)/(c*x^2+b*x+a)^(1/2),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c x^{2} + b x + a}{\left (e x + d\right )}^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(7/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(c*x^2 + b*x + a)*(e*x + d)^(7/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{c x^{2} + b x + a} \sqrt{e x + d}}{c e^{4} x^{6} +{\left (4 \, c d e^{3} + b e^{4}\right )} x^{5} + a d^{4} +{\left (6 \, c d^{2} e^{2} + 4 \, b d e^{3} + a e^{4}\right )} x^{4} + 2 \,{\left (2 \, c d^{3} e + 3 \, b d^{2} e^{2} + 2 \, a d e^{3}\right )} x^{3} +{\left (c d^{4} + 4 \, b d^{3} e + 6 \, a d^{2} e^{2}\right )} x^{2} +{\left (b d^{4} + 4 \, a d^{3} e\right )} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(7/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(c*x^2 + b*x + a)*sqrt(e*x + d)/(c*e^4*x^6 + (4*c*d*e^3 + b*e^4)*x^5 + a*d^4 + (6*c*d^2*e^2 + 4*b
*d*e^3 + a*e^4)*x^4 + 2*(2*c*d^3*e + 3*b*d^2*e^2 + 2*a*d*e^3)*x^3 + (c*d^4 + 4*b*d^3*e + 6*a*d^2*e^2)*x^2 + (b
*d^4 + 4*a*d^3*e)*x), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (d + e x\right )^{\frac{7}{2}} \sqrt{a + b x + c x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**(7/2)/(c*x**2+b*x+a)**(1/2),x)

[Out]

Integral(1/((d + e*x)**(7/2)*sqrt(a + b*x + c*x**2)), x)

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(7/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")

[Out]

Timed out